8x - 7x + 130 = 260 What value of x is the solution to the given equation?...

1. SIMPLIFY the left side by combining like terms

• Given equation: \(\mathrm{8x - 7x + 130 = 260}\)
• Since \(\mathrm{8x}\) and \(\mathrm{-7x}\) are like terms (both have variable x to the first power):
  \(\mathrm{8x - 7x = (8 - 7)x = 1x = x}\)
• This gives us: \(\mathrm{x + 130 = 260}\)

2. INFER the strategy to isolate the variable

• We need x by itself on one side of the equation
• Since 130 is being added to x, we need to subtract 130 from both sides

3. SIMPLIFY by performing the subtraction

• Subtract 130 from both sides:
  \(\mathrm{x + 130 - 130 = 260 - 130}\)
  \(\mathrm{x = 130}\)

Answer: 130

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skills: Students incorrectly combine the coefficients of like terms.

Instead of recognizing that \(\mathrm{8x - 7x = 1x = x}\), they might:

• Add the coefficients: \(\mathrm{8x - 7x = 15x}\) (thinking subtraction means addition)
• Keep terms separate and get confused about next steps
• Make basic arithmetic errors like \(\mathrm{8 - 7 = 2}\)

This leads to confusion and incorrect equations that don't lead to the right answer.

Second Most Common Error:

Poor INFER reasoning: Students don't recognize the proper inverse operation to isolate x.

When they reach \(\mathrm{x + 130 = 260}\), they might add 130 to both sides instead of subtracting, getting:

\(\mathrm{x + 130 + 130 = 260 + 130}\)
\(\mathrm{x + 260 = 390}\)

This leads them away from the correct solution and causes them to get stuck and guess.

The Bottom Line:

This problem tests fundamental algebraic manipulation skills. Success requires both accurate arithmetic when combining like terms and strategic thinking about how to isolate variables using inverse operations.